Introduction: Topics

• Assumptions on t-tests:
  • One-sample
    • Normality
  • Independent samples
    • Normality
    • Equal variance
  • Dependent samples
    • Normality
• Nonparametric alternatives
  • Independent medians (M_1-M_2)
  • Dependent medians (M_d)

Introduction: Normality Assumption

• All t-tests assume approximate normality of the data.

  • In the case of one-sample t-tests, the measure of interest must somewhat follow a normal distribution.

  • In the case of two-sample t-tests, the measure of interest in each group must somewhat follow a normal distribution.

• Note that a paired t-test is technically a one-sample t-test, so we will examine normality of the difference.

Normality Assumption: Quantile-Quantile Plots

• There are formal tests for normality (see article here), however, we will not use them.

  • Tests for normality are not well-endorsed by statisticians.

• Instead, we will assess normality using a quantile-quantile (Q-Q) plot.

• A Q-Q plot helps us visually check if our data follows a specific distribution (here, the normal).

  • It compares the quantiles of our sample data to the quantiles of a theoretical distribution (the normal).

• How do we read Q-Q plots?

  • Each dot represents one observation in our dataset.
  • If the data follow a normal distribution, we will observe that the dots fall roughly along a straight diagonal line.
  • We focus on the “middle” of the graph.

Normality Assumption: Quantile-Quantile Plots

Normality Assumption: Quantile-Quantile Plots

Normality Assumption: Quantile-Quantile Plots

Normality Assumption: Quantile-Quantile Plots

Normality Assumption: Independent Means

• Recall our example from last lecture: In the skies above Cloudsdale, Pegasus trainers believe that an average healthy Pegasus flaps its wings 50 flaps per minute when cruising. To see if today’s young Pegasi conform to that standard, a researcher samples 25 Pegasi at the Cloudsdale Training Grounds and measures each pony’s wing‐flap rate (in flaps/minute).
  

Normality Assumption: Independent Means

• Further, we performed a two-sample t-test to determine if the above target pegasi are eating 5 or more apples than the below target pegasi (\alpha=0.05).

wing_flap %>% independent_mean_HT(grouping = target,
                                  continuous = apples, 
                                  mu = 5, 
                                  alternative = "greater", 
                                  alpha = 0.05)

Two-sample t-test for two independent means and equal variance:
Null: H₀: μ₁ − μ₂ = 5
Alternative: H₁: μ₁ − μ₂ > 5
Test statistic: t(23) = 5.445
p-value: p < 0.001
Conclusion: Reject the null hypothesis (p = < 0.001 < α = 0.05)

• Are our results valid?

Normality Assumption: Independent Means (R)

• We will use the independent_qq() function from library(ssstats) to assess normality.

dataset_name %>% independent_qq(continuous = continuous_variable,
                                grouping = grouping_variable)

• This will provide the the Q-Q plots and the histograms for the two independent groups under consideration.

Normality Assumption: Independent Means

• Let’s now look at the normality assumption for our example.

• How should we change the code for our dataset?

dataset_name %>% independent_qq(continuous = continuous_variable,
                                grouping = grouping_variable)

Normality Assumption: Independent Means

• Let’s now look at the normality assumption for our example.

• How should we change the code for our dataset?

wing_flap %>% independent_qq(continuous = apples,
                             grouping = target)

Normality Assumption: Independent Means

• Running the code,

wing_flap %>% independent_qq(continuous = apples,
                             grouping = target)

Introduction: Variance Assumption

• In addition to normality, the two-sample t-test assumes equal variance between groups.

  • Homoskedastic: same variance
  • Heteroskedastic: different variances

• We can check this assumption and easily adjust if the assumption is broken.

• Graphical method: scatterplot of residuals

• Formal method: test for equal variances (Brown-Forsythe-Levine)

Variance Assumption: Residual Plot (R)

• We will use the plot_residuals() function from library(ssstats) to graphically assess the assumption of equal variance.

dataset_name %>% plot_residuals(continuous = continuous_variable,
                                grouping = grouping_variable)

• We should evaluate and compare the lengths of the resulting “lines”.

Variance Assumption: Residual Plot

• Let’s now look at the normality assumption for our example.

• How should we change the code for our dataset?

dataset_name %>% plot_residuals(continuous = continuous_variable,
                                grouping = grouping_variable)

Variance Assumption: Residual Plot

• Let’s now look at the normality assumption for our example.

• Our updated code:

wing_flap %>% plot_residuals(continuous = apples,
                             grouping = target)

Variance Assumption: Residual Plot

• Running the code,

wing_flap %>% plot_residuals(continuous = apples,
                             grouping = target)

Hypothesis Testing: Two or More Variances

• If we believe the assumption may be violated, we can test for equal variance using the Brown-Forsythe-Levine (BFL) test.

• This test is valid for more than two groups (read: we will see it again!)

• Hypotheses

  • H_0: \ \sigma^2_1 = \sigma^2_2 = ... = \sigma^2_k
  • H_1: at least one \sigma^2_i is different.

Hypothesis Testing: Two or More Variances

• Test Statistic:

F_0 = \frac{\sum_{i=1}^k n_i(\bar{z}_{i.}-\bar{z}_{..})^2/(k-1)}{\sum_{i=1}^k \sum_{j=1}^{n_i} (z_{ij}-\bar{z}_{i.})^2/(N-k)} \sim F_{\text{df}_{\text{num}}, \text{df}_{\text{den}}}

• where
  • n_i is the sample size of group i
  • \bar{z}_{i.} is the median of group i and \bar{z}_{..} is the grand median
  • k is the number of groups
  • N = \sum_{i=1}^k n_i is the overall sample size
  • \text{df}_{\text{num}} = k-1, \text{df}_{\text{den}} = (N-k).

Hypothesis Testing: Two or More Variances

• Note that the BFL is a one-tailed test, which is different than when we are testing means using the t distribution.

• p-value:

p = P\left[F_{\text{df}_{\text{num}}, \text{df}_{\text{den}}} \ge F_0\right]

Hypothesis Testing: Two or More Variances (R)

wing_flap %>% variances_HT(continuous = apples,
                           grouping = target)

Error in variances_HT(., continuous = apples, grouping = target): could not find function "variances_HT"

Variance Assumption: Independent Means

• example set up

Variance Assumption: Independent Means

• example set up with correct code

Variance Assumption: Independent Means

• code has been run

Variance Assumption: Independent Means

• HT results

Variance Assumption: Broken – Now What?

\text{df}=\frac{ \left( \frac{s^2_1}{n_1} + \frac{s_2^2}{n_2} \right)^2 }{ \frac{(s_1^2/n_1)^2}{n_1-1} + \frac{(s_2^2/n_2)^2}{n_2-1} }

Normality Assumption: Dependent Means

• normality assumption for dependent t

Normality Assumption: Dependent Means (R)

• syntax for qq

Normality Assumption: Dependent Means

• example set up

Normality Assumption: Dependent Means

• example set up with correct code

Normality Assumption: Dependent Means

• code has been run

Wrap Up: t-Test Assumptions

• Important note!!
  • I do not expect you to agree with my assessment of q-q plots!
  • What I do expect is that you know what to do after making your assessment.
    • Meet assumption \to use t-test
    • Do not meet assumption \to use nonparametric equivalent
• Next: nonparametrics!

Introduction: Nonparametrics

• The t-tests we have already learned are considered parametric methods.

  • There is a distributional assumption on the test.

• Nonparametric methods do not have distributional assumptions.

  • We typically transform the data to their ranks and then perform calculations.

• Why don’t we always use nonparametric methods?

  • They are often less efficient: a larger sample size is required to achieve the same probability of a Type I error.

  • They discard useful information :(

Introduction: Wilcoxon Rank Sum

• introduce W RS

Wilcoxon Rank Sum: Ranking Data

• In the nonparametric tests we will be learning, the data will be ranked.

• Let us first consider a simple example, x: \ 1, 7, 10, 2, 6, 8

• Our first step is to reorder the data: x: \ 1, 2, 6, 7, 8, 10

• Then, we replace with the ranks: R: \ 1, 2, 3, 4, 5, 6

Wilcoxon Rank Sum: Ranking Data

• What if all data values are not unique? We will assign the average rank for that group.

• For example, x: \ 9, 8, 8, 0, 3, 4, 4, 8

• Let’s reorder:x: \ 0, 3, 4, 4, 8, 8, 8, 9

• Rank ignoring ties:R: \ 1, 2, 3, 4, 5, 6, 7, 8

• Now, the final rank:R: \ 1, 2, 3.5, 3.5, 6, 6, 6, 8

Wilcoxon Rank Sum (R)

• syntax

Wilcoxon Rank Sum: Independent Medians

• example set up

Wilcoxon Rank Sum: Independent Medians

• example with correct code

Wilcoxon Rank Sum: Independent Medians

• code has been run

Wilcoxon Rank Sum: Independent Medians

• HT typeset